Correct Answer – Option 1 : tan 8/9

Given:

Sin (α – β) = 2√2/3 and cosec (α + β) = 2√2/3

Concept Used:

Basic concept of trigonometric ratio and identities

We know that

(α² – β²) = (α – β) × (α + β)

sin ^-1a × cosec ^-1a = 1

Calculation:

It is given that sin (α – β) = 2√2/3

∴ (α – β) = sin ^-1 2√2/3      —(1)

And cosec (α + β) = 2√2/3

∴ (α + β) = cosec ^-1 2√2/3      —(2)

By multiplying equation (1) and (2)

∴ (α – β) × (α + β) = sin ^-1 2√2/3 × cosec ^-1 2√2/3

⇒ (α² – β²) = 8/9

Now, tan (α² – β²) = tan 8/9

Hence, option (1) is correct